Question: EXERCISE 3 Let {Wt: t2 0} be a standard Brownian motion. We call the solution {Xt : t2 0} to the Langevin equation Xt =

EXERCISE 3 Let {Wt: t2 0} be a standard Brownian motion. We call the solution {Xt : t2 0} to the Langevin equation Xt = Wt+ Xsds the Ornstein-Uhlenbeck process. In particular, it is used for modeling short-term interest rates. 1. A first method to solve the Langevin equation: using the product formula for Ito processes, compute e tXt and deduce what a solution to the Langevin equation is. 2. A second method to solve the Langevin equation: using the Ito formula, prove that .= Jewsds te W . 3. Let Yt = et foerdWs. Using the Fubini theorem, prove that Yods = e esWads and deduce that Yt is a solution to the Langevin equation. 4. The law of the solution: Prove that, for any t 2 0, the law of Xt is a normal distribution N (0, ett e 2t - 1 )

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